# Instance-dependent uniform tail bounds for empirical processes

arXiv: https://arxiv.org/abs/2209.10053 · HTML: https://mlsanity.com/paper/2209.10053/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 40 references that are in the map.
- First posted on arXiv 21 Sep 2022.

## Builds on (most cited first, 3 of 3)

- [Learning with Square Loss: Localization through Offset Rademacher Complexity](https://mlsanity.com/paper/1502.06134/) (2015)
- [An optimal three-way stable and monotonic spectrum of bounds on quantiles: a spectrum of coherent measures of financial risk and economic inequality](https://mlsanity.com/paper/1310.6025/) (2013)
- [Multivariate mean estimation with direction-dependent accuracy](https://mlsanity.com/paper/2010.11921/) (2020)

## Abstract (arXiv)

We formulate a uniform tail bound for empirical processes indexed by a class of functions, in terms of the individual deviations of the functions rather than the worst-case deviation in the considered class. The tail bound is established by introducing an initial ``deflation'' step to the standard generic chaining argument. The resulting tail bound is the sum of the complexity of the ``deflated function class'' in terms of a generalization of Talagrand's γ functional, and the deviation of the function instance, both of which are formulated based on the natural seminorm induced by the corresponding Cram'er functions. Leveraging another less demanding natural seminorm, we also show similar bounds, though with implicit dependence on the sample size, in the more general case where finite exponential moments cannot be assumed. We also provide approximations of the tail bounds in terms of the more prevalent Orlicz norms or their ``incomplete'' versions under suitable moment conditions.